OMM 2004

Find all the prime number $p, q$ and r with $p < q < r$ , such that $25pq + r = 2004$ and $pqr + 1$ is a perfect square.

Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$ ) satisfy that $|a - b| \ge \frac{ab}{100} .$

Let $Z$ and $Y$ be the tangency points of the incircle of the triangle $ABC$ with the sides $AB$ and $CA$ , respectively. The parallel line to $Y Z$ through the midpoint $M$ of $BC$ , meets $CA$ in $N$ . Let $L$ be the point in $CA$ such that $NL = AB$ (and $L$ on the same side of $N$ than $A$ ). The line $ML$ meets $AB$ in $K$ . Prove that $KA = NC$ .

At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$ , then $A$ defeated $C$ . Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$ . How many teams played in the tournament? Find all possible answers.

Let $\omega_1$ and $\omega_2$ be two circles such that the center $O$ of $\omega_2$ lies in $\omega_1$ . Let $C$ and $D$ be the two intersection points of the circles. Let $A$ be a point on $\omega_1$ and let $B$ be a point on $\omega_2$ such that $AC$ is tangent to $\omega_2$ in C and BC is tangent to $\omega_1$ in $C$ . The line segment $AB$ meets $\omega_2$ again in $E$ and also meets $\omega_1$ again in F. The line $CE$ meets $\omega_1$ again in $G$ and the line $CF$ meets the line $GD$ in $H$ . Prove that the intersection point of $GO$ and $EH$ is the center of the circumcircle of the triangle $DEF$ .

What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$ , if the path does not cross the same place twice?.